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On a problem of Bernik, Kleinbock and Margulis. (English) Zbl 1268.11098
The author considers the system of inequalities $| P(x) | < \Psi_1(H(P)), \quad | P'(x) | < \Psi_2(H(P)),$ where $$P$$ is an integer polynomial, $$H(P)$$ is the maximum among the absolute values of the coefficients of $$P$$ and the $$\Psi_i$$ are real, positive functions. The set $${\mathcal P}_n(\Psi_1, \Psi_2)$$ consists of the numbers $$x \in [-1/2, 1/2]$$ for which this system of inequalities is satisfied for infinitely many integer polynomials $$P$$ of degree at most $$n$$.
Under various technical restrictions on the functions $$\Psi_1$$ and $$\Psi_2$$, the Lebesgue measure of the set $${\mathcal P}_n(\Psi_1, \Psi_2)$$ is calculated. With the technical assumptions ignored, the set is null or full according to the convergence or divergence of the series $$\sum_{h=1}^\infty h^{n-2} \Psi_1(h) \Psi_2(h)$$, a result very much like the classical Khintchine theorem. The paper is a first step towards establishing a Khintchine type theorem in this context, although there is still a long way to go due to the technical conditions. The proof uses Sprindzhuk’s method of essential and inessential domains along with more recent results.
##### MSC:
 11J83 Metric theory 11K60 Diophantine approximation in probabilistic number theory 11J13 Simultaneous homogeneous approximation, linear forms
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